As you observed,
$$ e^{ix} = e^{iy} \iff e^{i(x-y)} = 1 \iff \cos{(x-y)} = 1$$
(The condition $\sin{(x-y)} = 0$ is redundant because it follows from $\cos{(x-y)} = 1$ using $\cos^2\theta + \sin^2\theta = 1$.)
Now if $\cos\theta = 1$, then $\theta = 2n\pi$ pretty much by definition. One way of seeing this is that $\cos$ is periodic with period $2\pi$, and the only $\theta$ in $[0,2\pi)$ for which $\cos\theta = 1$ is $\theta=0$. Said differently, if $2\pi n$ is the multiple of $2\pi$ that is closest to $\theta$, then writing $\theta = 2\pi n + \phi$ where $-\pi < \phi \leq \pi$, we have $1 = \cos(\theta) = \cos(2\pi n + \phi) = \cos(\phi)$, so $\phi = 0$.
Alternatively, $\cos\theta$ is the projection on the x-axis of a point that has rotated by angle $\theta$ from its initial point $(1,0)$, so if $\cos\theta = 1$ (and therefore $\sin\theta = 0$), then the point has come back to $(1,0)$, so it must have made a number of full turns: if it made $n$ turns, then we say $\theta = 2\pi n$.
Those functions are much less used than before for one reason: the advent of electronic computers.
Before that, one had to rely either on tables or on slide rules. Tables were usually table of logarithms, and they included the logarithms of trigonometric functions as well. The trigonometric functions were then useful not only for geometric applications, but also to simplify algebraic calculations with logarithms.
For instance, to compute $\log\sqrt{a^2+b^2}$ when $\log a$ and $\log b$ are known, you could find $\theta$ such that $\log\tan\theta=\log\frac ba=\log b-\log a$, then $\log\sqrt{a^2+b^2}=\log a+\log\sqrt{1+\tan^2\theta}$ and $\log\sqrt{1+\tan^2\theta}=\log\frac{1}{\cos \theta}=-\log\cos\theta$. There are many similar formulas.
For geometric applications, sometimes versine and similar functions allow computing with greater precision while not adding too much computation. See for instance the haversine formula used to compute great circle distance (useful in navigation). The straightforward formula with arccosine has poor accuracy when the angle is small (the most common case), due to the fact that cosine is flat at $0$. However the formula with haversine is more accurate. To achieve the same, you would have to use $\sin^2(\theta/2)$ everywhere, which require more computations (but it's still reasonable with logarithms). Therefore, navigation table like Nories's nautical tables have an haversine table.
Note that tables of logarithms are usually accurate up to 5 digits (some larger tables had 7 digits, some very special ones had better precision but were difficult to use : more digits = much more space on paper). Slide rules have roughly 3 digits of precision.
All of this is rendered pretty useless with calculators, which have usually around 15 digits of precision and compute fast enough that we don't have to worry about speeding things up with extra functions.
Best Answer
As θ goes to zero, versin(θ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation, making separate tables for the latter convenient.
These days we walk around with devices capable of approximating the versine to a degree of accuracy that would in the past be considered an extraordinary degree of accuracy and capable of looking up tables the size of books for exact results.
That is the practical aspect, the other aspect is that it has fallen out of taste, and there is no accounting for taste so take it as you will.